SH - QQ topset in multiway 3b pot

Villain(s): MP = seems like a reg 3b 5.9% (2/34) assumed he was 3b'ing QQ+, Ako here only

Pre: 4b/call? Once I get the overcalls I would expect CO and BB to be capped at QQ and AK since KK+ almost always 4b here I think. I am acutally behind vs MP's 3b'ing range here and my default is to play QQ as a set mine, however given that there is more ded money to make up for the equity disadvantage would you lean more twoards 4b/broke here?

Flop: Considered x/raising however I was worried about the flop being checked through so decided to c-bet big since pot was multiway and due to the board texture being draw heavy. After MP folds to the c-bet I think I put him on too tight a range preflop. Once CO and BB just call I don't think that have made hands since I would expect sets and made straights to raise get it in unless they have AcKc/Kc9c/AcTc.

Turn: Given the above I think villains most likely have some sort of combo draw type hand, therefore decided to bomb the Turn in case they miss the river and I risk leaving money on the table. I think checking with the intention of x/raising would mostly result in a check behind by CO.

PokerStars - $0.05 NL (6 max) - Holdem - 6 players
Hand converted by PokerTracker 4

CO: $5.14 (VPIP: 23.08, PFR: 21.54, 3Bet Preflop: 7.41, Hands: 67)
BTN: $5.00 (VPIP: 22.00, PFR: 16.00, 3Bet Preflop: 8.70, Hands: 104)
SB: $9.67 (VPIP: 25.00, PFR: 20.31, 3Bet Preflop: 7.14, Hands: 65)
BB: $5.07 (VPIP: 20.19, PFR: 11.74, 3Bet Preflop: 4.44, Hands: 215)
Hero (UTG): $9.20
MP: $4.69 (VPIP: 25.00, PFR: 17.31, 3Bet Preflop: 4.55, Hands: 53)

SB posts SB $0.02, BB posts BB $0.05

Pre Flop: (pot: $0.07) Hero has Q♦ Q♠

Hero raises to $0.15, MP raises to $0.35, CO calls $0.35, fold, fold, BB calls $0.30, Hero calls $0.20

Flop: ($1.42, 4 players) J♣ T♥ Q♣
BB checks, Hero bets $0.91, fold, CO calls $0.91, BB calls $0.91

Turn: ($4.15, 3 players) 6♥
BB checks, Hero bets $3.98, CO calls $3.88 and is all-in, fold

River: ($11.91, 2 players) 4♥